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Let $X$ and $Y$ be any two graphs of order $n$. The friends-and-strangersgraph $\mathsf{FS}(X,Y)$ of $X$ and $Y$ is a graph with vertex set consistingof all bijections $\sigma :V(X) \mapsto V(Y)$, in which two bijections$\sigma$, $\sigma'$ are adjacent if and only if they differ precisely on twoadjacent vertices of $X$, and the corresponding mappings are adjacent in $Y$.The most fundamental question that one can ask about thesefriends-and-strangers graphs is whether or not they are connected. Let$\mathsf{Lollipop}_{n-k,k}$ be a lollipop graph of order $n$ obtained byidentifying one end of a path of order $n-k+1$ with a vertex of a completegraph of order $k$. Defant and Kravitz started to study the connectedness of$\mathsf{FS}(\mathsf{Lollipop}_{n-k,k},Y)$. In this paper, we give a sufficientand necessary condition for $\mathsf{FS}(\mathsf{Lollipop}_{n-k,k},Y)$ to beconnected for all $2\leq k\leq n$.

The connectedness of the friends-and-strangers graph of lollipop graphs and others